The radii of two concentric circles are 34 cm and 50 cm. A and D are the points on larger circle and B and C are points on smaller circle. If ABCD is a straight line and BC = 32 cm, then what is the length of AD?

80 cm

BC = 32 cm

BP = \(\frac{32}{2}\) = 16 cm

In triangle OBP

Apply pythagoras theorem

\( {OB }^{2 } \) = \( {BP }^{2 } \) + \( {OP }^{2 } \)

\( {34 }^{2 } \) = \( {OP }^{2 } \) + \( {16 }^{2 } \)

\( {OP }^{2 } \) = 1156 - 256

\( {OB }^{2 } \) = 900

OP = 30 cm

In triangle OPD

\( {OD }^{2 } \) = \( {OP }^{2 } \) + \( {PD }^{2 } \)

= \( {50 }^{2 } \) = \( {30 }^{2 } \) + \( {PD }^{2 } \)

= \( {PD }^{2 } \) = 2500 - 900

= \( {PD }^{2 } \) = 1600

= PD = 40 cm

= AD = 2 x PD = 2 x 40 = 80 cm

Therefore, the length of AD is 80 cm.